---
title: "Small o vs small omega"
subject: "Data Structure and Algorithm"
chapter: 9
key: "semester_4/data_structure_and_algorithm:small-o-vs-small-omega"
source: https://www.bctengineer.com/q/semester_4/data_structure_and_algorithm/small-o-vs-small-omega
site: BCT Engineering
---

# Small o vs small omega

Data Structure and Algorithm · Chapter 9: Growth Functions

- Asked in: 2070 Magh, 2 marks
- Video: [Little oh and Little Omega | Asymptotic Notations | Algorithm Analysis | Data Structures](https://www.youtube.com/watch?v=_1DPggw0ixg) — MathsInDepth (Decoding Science)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/data_structure_and_algorithm/small-o-vs-small-omega
- Whole chapter: https://www.bctengineer.com/md/semester_4/data_structure_and_algorithm/chapter-9

## Question

Differentiate between small o and small ω notations.

## Short answer

Small o: f grows strictly slower than g (lim f/g = 0, strict upper bound). Small ω: f grows strictly faster (lim f/g = ∞, strict lower bound). Example: n = o(n²) but n² ≠ o(n²); n² = ω(n) but n² ≠ ω(n²).

## Answer

| Point | Small o | Small ω |
|---|---|---|
| Meaning | f grows strictly slower than g | f grows strictly faster than g |
| Definition | f(n) < c·g(n) for **every** c > 0, for n ≥ n₀ | f(n) > c·g(n) for **every** c > 0, for n ≥ n₀ |
| Limit | lim f(n)/g(n) = 0 | lim f(n)/g(n) = ∞ |
| Bound type | Strict upper bound | Strict lower bound |
| Relation to big forms | o ⊂ O (not tight) | ω ⊂ Ω (not tight) |
| Example | n = o(n²), but n² ≠ o(n²) | n² = ω(n), but n² ≠ ω(n²) |
| Analogy | f < g | f > g |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does f = o(g) mean?
   A. f grows strictly slower than g
   B. f grows strictly faster than g
   C. f equals g
   D. f is at most g, possibly equal
   Answer: A. lim f/g = 0. (Key point: Small o)

2. What limit condition defines f = omega(g)?
   A. lim f(n)/g(n) = 0
   B. lim f(n)/g(n) = infinity
   C. lim f(n)/g(n) = 1
   D. lim f(n)/g(n) = 5
   Answer: B. f dominates g strictly. (Key point: Small omega limit)

3. Which statement is true?
   A. n^2 = o(n^2)
   B. n^2 = o(n)
   C. n = o(n^2) but n^2 is not o(n^2)
   D. n = omega(n^2)
   Answer: C. Small o is strict, so a function is never o of itself. (Key point: Example)

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